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Curved Lorentzian potential wave identity

Statement

Let $(M, g, N, o)$ be an oriented, time-oriented smooth Lorentzian four-manifold of signature $(-, +, +, +)$ with the metric-selected connection $\nabla$ of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Use the curvature operator, Ricci contraction, and sign convention of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Let $A$ be a real one-form whose coefficients in every smooth chart are $C^{3}$ . On such coefficients, use the covariant-derivative component formula $\nabla_{a} A_{b} : = \partial_{a} A_{b} - Γ^{c}_{a b} A_{c}$ and define $d A$ as the $C^{2}$ two-form with chart components $(d A)_{a b} : = \partial_{a} A_{b} - \partial_{b} A_{a}$ ; for smooth $A$ this agrees with the exterior derivative of Lorentzian exterior calculus and Hodge star. The transformation law for these antisymmetrized components follows by the chain rule, since the second-derivative chart-transition terms cancel by equality of mixed partials. Define $(□ A)_{b} : = g^{a c} \nabla_{a} \nabla_{c} A_{b}$ , $(div A) : = \nabla^{a} A_{a}$ , and $(P A)_{b} : = □ A_{b} - {Ric}_{b}^{c} A_{c}$ . Then

$\nabla^{a} (d A)_{a b} = (P A)_{b} - \nabla_{b} (div A) .$

The same identity holds for a complex one-form after complex-linear extension. The identity is local and uses no field equation, gauge condition, boundary condition, or physical assumption.

Facts & Assumptions

Given: A smooth oriented, time-oriented Lorentzian four-manifold with metric-compatible torsion-free connection, a real $C^{3}$ one-form $A$ , and the curvature convention in the Statement.

[F1]

Oriented Lorentzian fields, observers and coordinate conventions supplies the smooth oriented Lorentzian four-manifold and the metric-selected connection convention; Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields states that this connection is torsion-free, metric-compatible, and extends to tensor fields.

[F2]

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors defines $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ , $R_{a b c d} = g_{a e} R^{e}_{b c d}$ , and ${Ric}_{b d} = g^{a c} R_{a b c d}$ .

[F3]

The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors proves $R_{a b c d} = - R_{a b d c}$ , $R_{a b c d} = R_{c d a b}$ , and ${Ric}_{a b} = {Ric}_{b a}$ for this curvature convention.

[F4]

Dual connection defines the induced covector connection by $(\nabla_{X} ω) (Z) = X (ω (Z)) - ω (\nabla_{X} Z)$ .

[F5]

Lorentzian exterior calculus and Hodge star supplies $(d A)_{a b} = 2 \partial_{[a} A_{b]}$ and the real-linearity of $d$ .

[F6]

Clairaut--Schwarz theorem for continuous second partial derivatives states that continuous second partial derivatives commute; it supplies the cancellation of the second-derivative chart-transition terms for the stated $C^{3}$ extension of $d$ .

Proof

1.1F1F5F6algebra

In any chart the stated extension has $(d A)_{a b} = \partial_{a} A_{b} - \partial_{b} A_{a}$ and $\nabla_{a} A_{b} = \partial_{a} A_{b} - Γ^{c}_{a b} A_{c}$ . Under a chart change, differentiating $A_{b}^{'} = (\partial x^{c} / \partial y^{b}) A_{c}$ and antisymmetrizing cancels the second-derivative transition terms by [F6], so these components define a two-form. Torsion freedom gives $Γ^{c}_{a b} = Γ^{c}_{b a}$ , so $(d A)_{a b} = \nabla_{a} A_{b} - \nabla_{b} A_{a}$ . Therefore $\nabla^{a} (d A)_{a b} = □ A_{b} - \nabla^{a} \nabla_{b} A_{a}$ .

1.2F2F4algebra

For vector fields $X, Y, Z$ , expand the dual-connection rule [F4] twice. The scalar-derivative terms cancel by $X (Y f) - Y (X f) = [X, Y] f$ , leaving $([\nabla_{X}, \nabla_{Y}] ω - \nabla_{[X, Y]} ω) (Z) = - ω (R (X, Y) Z)$ for every one-form $ω$ . Since coordinate fields commute, $[\nabla_{c}, \nabla_{b}] A_{a} = - R^{d}_{a c b} A_{d}$ .

2.1F2F3step 1.2algebra

Contract the last identity with $g^{a c}$ . By metric compatibility, $g^{a c} [\nabla_{c}, \nabla_{b}] A_{a} = \nabla^{a} \nabla_{b} A_{a} - \nabla_{b} \nabla^{a} A_{a}$ . By [F3] and the Ricci definition in [F2], $g^{a c} R^{d}_{a c b} = g^{d e} g^{a c} R_{e a c b} = g^{d e} g^{a c} R_{b c a e} = - g^{d e} g^{a c} R_{a e c b} = - {Ric}_{b}^{d}$ ; thus $\nabla^{a} \nabla_{b} A_{a} = \nabla_{b} (\nabla^{a} A_{a}) + {Ric}_{b}^{d} A_{d}$ .

3.1givenstep 1.1step 2.1algebra∎

Substituting step 2.1 into step 1.1 gives $\nabla^{a} (d A)_{a b} = □ A_{b} - \nabla_{b} (\nabla^{a} A_{a}) - {Ric}_{b}^{d} A_{d} = (P A)_{b} - \nabla_{b} (div A)$ by the definition of $P$ . All identities are pointwise tensor identities, so they hold in every chart; complex-linear extension follows by applying the real identity to the real and imaginary parts separately.

Curved Lorentzian potential wave identity

Statement

Facts & Assumptions

Proof

Depends on

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